# Mathematical Statistics and Data Analysis - Solutions

### Chapter 7, Survey Sampling

#### (a)

We need $\, \Exp(\hat \beta) = \beta \,$. Since $\, \hat \beta = w_1 \hat X_1 + w_2 \hat X_2 + w_3 \hat X_3 \,$, we have $\, \Exp(\hat \beta) = (w_1 + w_2 + w_3) \alpha + (w_1 + 2 w_2 + 3 w_3) \beta \,$(Using $\Exp(\bar X_i) = \mu(t_i) = \alpha + \beta t_i$) should be equal to $\, \beta \,$. Thus $\, w_1 + w_2 + w_3 = 0 \,$ and $\, w_1 + 2 w_2 + 3 w_3 = 1 \,$.

#### (b)

TODO

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